Related papers: The stochastic logarithmic Schr\"odinger equation
We prove global existence of small solutions to the initial value problem for a class of cubic derivative nonlinear Schr\"odinger systems with the masses satisfying suitable non-resonance relations. The large-time asymptotics of the…
This work is concerned with existence and uniqueness of solutions to the reflection problem for linear parabolic equation with multiplicative Gaussian noise.
Within the framework of stochastic Schroedinger equations, we show that the correspondence between statevector equations and ensemble equations is infinitely many to one, and we discuss the consequences. We also generalize the results of…
In this paper, we study the Schr\"odinger equation associated with the Weinstein operators and we prove the existence and uniqueness of global solutions to Schr\"odinger-Weinstein equations in\\ $C\left(\left(-T_{\min }, T_{\max }\right) ;…
In this article, we study the stochastic wave equation on the entire space $\mathbb{R}^d$, driven by a space-time L\'evy white noise with possibly infinite variance (such as the $\alpha$-stable L\'evy noise). In this equation, the noise is…
We prove the global existence of solution to the small data mass critical stochastic nonlinear Schr\"{o}dinger equation in $d=1$. We further show the stability of the solution under perturbation of initial data. Our construction starts with…
In this paper, we study the defocusing nonlinear Schr\"{o}dinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
We address the global existence of solutions to the stochastic Navier-Stokes equations with multiplicative noise and with initial data in $H^{1/2}(\mathbb{T}^{3})$. We prove that the solution exists globally in time with probability…
We study a system of two coupled nonlinear Schr\"{o}dinger equations, where one equation includes gain and the other one includes losses. Strengths of the gain and the loss are equal, i.e., the resulting system is parity-time (${\cal PT}$)…
In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the $I$-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we…
In this paper we study the stochastic inhomogeneous incompressible Euler equations in the whole space $\RR^3$. We prove the existence and pathwise uniqueness of local solutions with both additive and multiplicative stochastic noise. Our…
We present a new parallel numerical method for solving the non-stationary Schr\"odinger equation with linear nonlocal condition and time-dependent potential which does not commute with the stationary part of the Hamiltonian. The given…
The non-relativistic Schrodinger equation with the linear and Coulomb potentials is solved numerically in configuration space using the relaxation method. The numerical method presented in this paper is a plain explicit Schrodinger solver…
We present a regularized finite difference method for the logarithmic Schr\"odinger equation (LogSE) and establish its error bound. Due to the blow-up of the logarithmic nonlinearity, i.e. $\ln \rho\to -\infty$ when $\rho\rightarrow 0^+$…
In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schr\"odinger operator and its fractional pseudo-relativistic counterpart, both arising in quantum models with nonlocal and relativistic…
For almost 75 years, the general solution for the Schr\"odinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of a solution in the case of a singular…
In this article we are concerned with the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -{\epsilon}^2\Delta u+ V(x)u=u \log u^2, & \mbox{in} \,\, \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\…
We study the nonlinear Schr\"odinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially…
We use Strichartz estimates with rough potentials like the spatial white noise on the 2 \ dimensional torus to prove global well-posedness of the multiplicative stochastic NLS with general integer powers in both the energy and strong regime…